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Up to a polynomial.

What if the polynomial is of order 1000 or higher?

I understand asymptotics and the convention that polynomial algorithms are "efficient". However, it might turn out that P=NP but the polynomial is so big that the found algorithm is impractical for normal sized instances.

That's certainly a logical possibility. In practice, low-degree polynomial solutions have been developed for most important problems in P, even if the first known polynomial solution had a high degree.

What would be even more entertaining, is a non-constructive proof of NP=P.

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